This paper introduces a new algorithm to improve the accuracy of numerical phase-averaging in oscillatory, multiscale, differential equations. Phase-averaging is a timestepping method which averages a mapped variable to remove highly oscillatory linear terms from the differential equation. This retains the main contribution of fast waves on the low frequencies without explicitly resolving the rapid oscillations. However, this comes at the cost of introducing an averaging error. To offset this, we propose a modified mapping that includes a mean correction term encoding an average measure of the nonlinear interactions. This mapping was introduced in Tao (2019) for weak nonlinearity and relied on classical time-averaging, which leaves only the zero frequencies. Our algorithm instead considers mean corrected phase-averaging when 1) the nonlinearity is not weak but the linear oscillations are fast and 2) finite averaging windows are applied via a smooth kernel, which has the advantage of retaining low frequencies whilst still eliminating the fastest oscillations. In particular, we introduce a local mean correction that combines the concepts of a mean correction and finite averaging; this retains low-frequency components in the mean correction that are removed with classical time-averaging. We show that the new timestepping algorithm reduces phase errors in the mapped variable for the swinging spring ODE in various dynamical configurations. We also show accuracy improvements with a local mean correction compared to standard phase-averaging in the one-dimensional rotating shallow water equations, a useful test case for weather and climate applications.

翻译：本文提出一种新算法，用以提高振荡多尺度微分方程中数值相位平均的精度。相位平均是一种时间步进方法，通过对映射变量进行平均来消除微分方程中的高频线性项。这种方法在不显式解析快速振荡的情况下，保留了快波对低频分量的主要贡献。然而，这会引入平均误差。为抵消此误差，我们提出一种改进的映射方法，其中包含一个编码非线性相互作用平均效应的均值校正项。该映射由Tao（2019）针对弱非线性情形引入，并依赖于经典时间平均方法，该方法仅保留零频分量。我们的算法则考虑在以下两种情况下应用均值校正相位平均：1）非线性非弱但线性振荡快速；2）通过光滑核函数应用有限平均窗口，其优势在于保留低频分量同时仍能消除最快振荡。特别地，我们引入了一种结合均值校正与有限平均概念的局部均值校正方法；该方法在均值校正中保留了被经典时间平均所去除的低频分量。我们证明，对于摆动弹簧常微分方程在不同动力学构型下，新的时间步进算法能减少映射变量的相位误差。同时，在一维旋转浅水方程（天气与气候应用中的重要测试案例）中，我们展示了局部均值校正相比标准相位平均方法的精度提升。